Higgs Quadruplet Impact on $W$ Mass Shift, Dark Matter, and LHC Signatures

The addition of a Higgs quadruplet to the standard model (SM) of quarks and leptons would shift the $W$ boson mass upward. It could also facilitate the production of dark matter through the conventional thermal freeze-out scenario via Yukawa interaction with the Higgs quadruplet or freeze-in production from the decay of SM Higgs. We investigate the same-sign lepton smoking gun signature of the double-charged scalar component of the Quadruplet Higgs at the LHC.